Theory of Computation Lecture Notes. Problems and Algorithms. Class Information


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1 Theory of Computation Lecture Notes Prof. YuhDauh Lyuu Dept. Computer Science & Information Engineering and Department of Finance National Taiwan University Problems and Algorithms c 2004 Prof. YuhDauh Lyuu, National Taiwan University Page 1 c 2004 Prof. YuhDauh Lyuu, National Taiwan University Page 3 Class Information Papadimitriou. Computational Complexity. 2nd printing. AddisonWesley Check I have never done anything useful. Godfrey Harold Hardy ( ), A Mathematician s Apology (1940) for lecture notes. c 2004 Prof. YuhDauh Lyuu, National Taiwan University Page 2 c 2004 Prof. YuhDauh Lyuu, National Taiwan University Page 4
2 Tractability and intractability What This Course Is All About Computability: What can be computed? What is computation anyway? There are welldefined problems that cannot be computed. In fact, most problems cannot be computed. Polynomial in terms of the input size n defines tractability. n, n log n, n 2, n 90. Time, space, circuit size, number of random bits, etc. It results in a fruitful and practical theory of complexity. Few practical, tractable problems require a large degree. Exponentialtime or superpolynomialtime algorithms are usually impractical. n log n, 2 n, 2 n, n! 2πn (n/e) n. c 2004 Prof. YuhDauh Lyuu, National Taiwan University Page 5 c 2004 Prof. YuhDauh Lyuu, National Taiwan University Page 7 What This Course Is All About (concluded) Complexity: What is a computable problem s inherent complexity? Some computable problems require at least exponential time and/or space; they are intractable. Some practical problems require superpolynomial resources unless certain conjectures are disproved. Other resource limits besides time and space? Program size, circuit size (growth), number of random bits, etc. Growth of Factorials n n! n n! , ,628, ,916, ,001, ,227,020, ,178,291, ,307,674,368, ,922,789,888,000 c 2004 Prof. YuhDauh Lyuu, National Taiwan University Page 6 c 2004 Prof. YuhDauh Lyuu, National Taiwan University Page 8
3 Turing Machines a A Turing machine (TM) is a quadruple M = (K, Σ, δ, s). K is a finite set of states. s K is the initial state. Turing Machines Σ is a finite set of symbols (disjoint from K). Σ includes (blank) and (first symbol). δ : K Σ (K {h, yes, no }) Σ {,, } is a transition function. (left), (right), and (stay) signify cursor movements. a Turing (1936). c 2004 Prof. YuhDauh Lyuu, National Taiwan University Page 9 c 2004 Prof. YuhDauh Lyuu, National Taiwan University Page 11 What Is Computation? A TM Schema That can be coded in an algorithm. An algorithm is a detailed stepbystep method for solving a problem. The Euclidean algorithm for the greatest common divisor is an algorithm. Let s be the least upper bound of compact set A is not an algorithm. Let s be a smallest element of a finitesized array can be solved by an algorithm δ c 2004 Prof. YuhDauh Lyuu, National Taiwan University Page 10 c 2004 Prof. YuhDauh Lyuu, National Taiwan University Page 12
4 The Operations of TMs Physical Interpretations The tape: computer memory and registers. δ: program. K: instruction numbers. s: main() in C. Σ: alphabet much like the ASCII code. Initially the state is s. The string on the tape is initialized to a, followed by a finitelength string x (Σ { }). x is the input of the TM. The input must not contain s (why?)! The cursor is pointing to the first symbol, always a. The TM takes each step according to δ. The cursor may overwrite to make the string longer during the computation. c 2004 Prof. YuhDauh Lyuu, National Taiwan University Page 13 c 2004 Prof. YuhDauh Lyuu, National Taiwan University Page 15 More about δ The program has the halting state (h), the accepting state ( yes ), and the rejecting state ( no ). Given current state q K and current symbol σ Σ, δ(q, σ) = (p, ρ, D). It specifies the next state p, the symbol ρ to be written over σ, and the direction D the cursor will move afterwards. We require δ(q, ) = (p,, ) so that the cursor never falls off the left end of the string. The Halting of a TM A TM M may halt in three cases. yes : M accepts its input x, and M(x) = yes. no : M rejects its input x, and M(x) = no. h: M(x) = y, where the string consists of a, followed by a finite string y, whose last symbol is not, followed by a string of s. y is the output of the computation. y may be empty denoted by ɛ. If M never halts on x, then write M(x) =. c 2004 Prof. YuhDauh Lyuu, National Taiwan University Page 14 c 2004 Prof. YuhDauh Lyuu, National Taiwan University Page 16
5 Why TMs? Because of the simplicity of the TM, the model has the advantage when it comes to complexity issues. One can develop a complexity theory based on C++ or Java, say. But the added complexity does not yield additional fundamental insights. We will describe TMs in pseudocode. Configurations (concluded) A configuration is a triple (q, w, u): q K. w Σ is the string to the left of the cursor (inclusive). u Σ is the string to the right of the cursor. Note that (w, u) describes both the string and the cursor position. c 2004 Prof. YuhDauh Lyuu, National Taiwan University Page 17 c 2004 Prof. YuhDauh Lyuu, National Taiwan University Page 19 The Concept of Configuration A configuration is a complete description of the current state of the computation. The specification of a configuration is sufficient for the computation to continue as if it had not been stopped. What does your PC save before it sleeps? Enough for it to resume work later w = u = c 2004 Prof. YuhDauh Lyuu, National Taiwan University Page 18 c 2004 Prof. YuhDauh Lyuu, National Taiwan University Page 20
6 Fix a TM M. Yielding Configuration (q,w, u) yields configuration (q, w, u ) in one step, (q, w, u) M (q, w, u ), if a step of M from configuration (q, w, u) results in configuration (q, w, u ). (q,w, u) Mk (q, w, u ): Configuration (q,w, u) yields configuration (q, w, u ) in k N steps. (q,w, u) M (q, w, u ): Configuration (q,w, u) yields configuration (q, w, u ). Palindromes A string is a palindrome if it reads the same forwards and backwards (e.g., ). A TM program can be written to recognize palindromes: It matches the first character with the last character. It matches the second character with the next to last character, etc. (see next page). yes for palindromes and no for nonpalindromes. This program takes O(n 2 ) steps. Can we do better? c 2004 Prof. YuhDauh Lyuu, National Taiwan University Page 21 c 2004 Prof. YuhDauh Lyuu, National Taiwan University Page 23 Example: How to Insert a Symbol We want to compute f(x) = ax. The TM moves the last symbol of x to the right by one position. It then moves the next to last symbol to the right, and so on. The TM finally writes a in the first position The total number of steps is O(n), where n is the length of x. c 2004 Prof. YuhDauh Lyuu, National Taiwan University Page 22 c 2004 Prof. YuhDauh Lyuu, National Taiwan University Page 24
7 Decidability and Recursive Languages Let L (Σ { }) be a language, i.e., a set of strings of symbols with a finite length. For example, {0, 01, 10, 210, 1010,...}. Let M be a TM such that for any string x: If x L, then M(x) = yes. If x L, then M(x) = no. We say M decides L. If L is decided by some TM, then L is recursive. Palindromes over {0, 1} are recursive. Acceptability and Recursively Enumerable Languages (concluded) If L is accepted by some TM, then L is a recursively enumerable language. A recursively enumerable language can be generated by a TM, thus the name. That is, there is an algorithm such that for every x L, it will be printed out eventually. c 2004 Prof. YuhDauh Lyuu, National Taiwan University Page 25 c 2004 Prof. YuhDauh Lyuu, National Taiwan University Page 27 Recursive and Recursively Enumerable Languages Acceptability and Recursively Enumerable Languages Let L (Σ { }) be a language. Let M be a TM such that for any string x: If x L, then M(x) = yes. If x L, then M(x) =. We say M accepts L. Proposition 1 If L is recursive, then it is recursively enumerable. We need to design a TM that accepts L. Let TM M decide L. We next modify M s program to obtain M that accepts L. M is identical to M except that when M is about to halt with a no state, M goes into an infinite loop. M accepts L. c 2004 Prof. YuhDauh Lyuu, National Taiwan University Page 26 c 2004 Prof. YuhDauh Lyuu, National Taiwan University Page 28
8 TuringComputable Functions Let f : (Σ { }) Σ. Optimization problems, root finding problems, etc. Let M be a TM with alphabet Σ. M computes f if for any string x (Σ { }), M(x) = f(x). We call f a recursive function a if such an M exists. a Gödel (1931). Extended Church s Thesis All reasonably succinct encodings of problems are polynomially related. Representations of a graph as an adjacency matrix and as a linked list are both succinct. The unary representation of numbers is not succinct. The binary representation of numbers is succinct vs All numbers for TMs will be binary from now on. c 2004 Prof. YuhDauh Lyuu, National Taiwan University Page 29 c 2004 Prof. YuhDauh Lyuu, National Taiwan University Page 31 Church s Thesis or the ChurchTuring Thesis What is computable is Turingcomputable; TMs are algorithms (Kleene 1953). Many other computation models have been proposed. Recursive function (Gödel), λ calculus (Church), formal language (Post), assembly languagelike RAM (Shepherdson & Sturgis), boolean circuits (Shannon), extensions of the Turing machine (more strings, twodimensional strings, and so on), etc. All have been proved to be equivalent. No intuitively computable problems have been shown not to be Turingcomputable (yet). Turing Machines with Multiple Strings A kstring Turing machine (TM) is a quadruple M = (K, Σ, δ, s). K, Σ, s are as before. δ : K Σ k (K {h, yes, no }) (Σ {,, }) k. All strings start with a. The first string contains the input. Decidability and acceptability are the same as before. When TMs compute functions, the output is on the last (kth) string. c 2004 Prof. YuhDauh Lyuu, National Taiwan University Page 30 c 2004 Prof. YuhDauh Lyuu, National Taiwan University Page 32
9 A 2String TM δ δ ababbaabbaabbaabbaba ababbaabbaabbaabbaba c 2004 Prof. YuhDauh Lyuu, National Taiwan University Page 33 c 2004 Prof. YuhDauh Lyuu, National Taiwan University Page 35 palindrome Revisited A 2string TM can decide palindrome in O(n) steps. It copies the input to the second string. The cursor of the first string is positioned at the first symbol of the input. The cursor of the second string is positioned at the last symbol of the input. The two cursors are then moved in opposite directions until the ends are reached. The machine accepts if and only if the symbols under the two cursors are identical at all steps. Configurations and Yielding The concept of configuration and yielding is the same as before except that a configuration is a (2k + 1)triple w i u i is the ith string. (q, w 1, u 1, w 2, u 2,...,w k, u k ). The ith cursor is reading the last symbol of w i. Recall that is each w i s first symbol. The kstring TM s initial configuration is 2k {}}{ (s,, x,, ɛ,, ɛ,...,, ɛ). c 2004 Prof. YuhDauh Lyuu, National Taiwan University Page 34 c 2004 Prof. YuhDauh Lyuu, National Taiwan University Page 36
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